Do you find yourself spending endless hours waiting at the airport? Here’s what math says about the perfect time to arrive for your next flight

If you've never missed a flight, you're probably spending too much time in airports. It's a counterintuitive idea—why would anyone want to risk missing a plane?—but it's got some logical thinking behind it, first from Nobel Prize-winning economist George Stigler, who famously touted the idea, and more recently, from math professor Jordan Ellenberg, who breaks down the idea in his new book How Not to Be Wrong: The Power of Mathematical Thinking.

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The question of when to arrive at the airport, Ellenberg argues, boils down to the basic question of utility, an economic concept used to measure the benefits and costs of something to someone. Utility can be either positive or negative: nice things, like puppies and cake, are (for most people) positive, while bad things, like illness or jury duty, are negative. There's positive utility, for some people, in not spending an excessive amount of time at the airport. There's also positive utility in not missing your flight. When you should arrive at the airport, Ellenberg says, is at the time that optimizes your personal utility: the time that minimizes both your chance of missing the flight and your chance of having so much excess time that you're forced to take your tenth lap around the airport waiting area.

Ellenberg breaks this down in mathematical terms by using a measurement of utility called utils. Let's say that an hour of your time, to you, is worth one util. Arriving at the airport two hours early wastes you two hours of time, so you lose two utils. But missing a plane is more annoying to you than wasting time at an airport—maybe it's six times more annoying than an hour of wasted time, so missing your flight costs you six utils. Ellenberg uses this quantification to assess the utility of three different scenarios:

Option 1: arrive two hours before flight, miss flight two percent of the time

Option 2: arrive one and a half hours before flight, miss flight five percent of the time

Option 3: arrive one hour before flight, miss flight fifteen percent of the time

Using util values for time, you can figure out which scenario affords you the most positive utility. In the first scenario, two hours of your time equals -2 utils (negative because it's a loss to be wasting time), but the chance of missing the flight two percent of the time has to be accounted for as well (-6 utils times the two percent chance of that happening). When the two are added together, the utility for the first scenario lands at -2.12 utils. For option two, your utility ends up being -1.8 (-1.5 utils plus -6 times five percent) and for option three, your utility is -1.9 utils. So from a mathematical standpoint, your best bet would be to arrive one and a half hours before your flight.

Of course, the above example assigns an esoteric util rate to an hour. You might find waiting at the airport more unbearable than the average person—maybe the thought of spending another minute flipping through magazines at an airport kiosk drives you crazy enough that it would take 10 utils. Or maybe the idea of missing a flight is a nightmare for you, and it would cost you 50 utils. That's going to change the equation, and affect your perfect time for an airport arrival. "What is always the case is that the best point is somewhere in between the two extremes, which means, in particular, that when you take the optimal time, whatever it is, your chance of missing the plane is not zero," Ellenberg explains. "It’s probably pretty small for most people, but it’s not literally zero."

And since that probability isn't zero, that means that someday, if you take enough flights, sooner or later you're going to miss one. If you fly frequently, and haven't missed a plane, then you're probably wasting too much time at the airport—and draining some utility along with your time.

Theoretically, that makes sense, but it doesn't necessarily answer the question of when to arrive for your flight if you want to minimize wasted time and your chance of missing your plane. The Transportation Security Administration (TSA) doesn't have an official recommendation when it comes to arrival time—it says that many factors affect the wait time at an airport—but arriving two hours before a flight is scheduled to take off has become a rule of thumb for many airport travelers. Various airlines have their own suggestions—Delta, for instance, has compiled this handy chart for travelers, which tells them the minimum check in time for popular domestic airports and United and American both offers their passengers something similar. Fliers who really want to take the guesswork out of their arrival time can even track the security wait times at various airports, either by applying for TSA's new Pre-Check program, which allows prescreened passengers (who have undergone a background check and been fingerprinted) to breeze through special security lanes, or by downloading the TSA mobile app, which allows users to report the wait time at their airport, creating a kind of wait time database.

But there is one tangible way to gage whether or not you can risk arriving to the airport a little later than normal: check what time your flight leaves, and decide whether or not the airport is going to be particularly busy during that time. If you're flying during outside of "peak" travel hours, the airports are likely to be less busy, meaning you can push your arrival time without getting caught in security and check-in lines. When are "peak" times to avoid (or maybe get to the airport a little earlier)? Los Angeles International Airport lists them as times that coincide with international arrivals—between 11 a.m. and 2 p.m. and 8 to 11 p.m. daily. Continental Airlines suggests arriving early if you're flying on a Friday or Sunday evening, Monday morning or between the hours of 6:30-9:30 a.m., 11 a.m-2 p.m. and 3:30-7:30 p.m. So if your flight leaves Tuesday at 3 p.m., you might be better off waiting for peak traffic to dissipate around 2 p.m., rather than arriving two hours before your flight just to stand in longer lines.

Still, if you find yourself on the wrong end of Ellenberg's spectrum—either with too much time or a missed flight on your hands—there are better airports to be stuck in than others. Changi Airport in Singapore features a butterfly and orchid garden, while Amsterdam's Schipol offers travelers a chance to see the sights of the famous Rijksmuseum within the airport's walls (they also have a library for passengers that opened in 2010). If you're flying JetBlue out of New York's JFK Airport, you might want to go ahead and completely disregard Ellenberg's suggestion: opened in 2008, JetBlue's Terminal 5 features free WiFi throughout the entire terminal, along with 29 places to shop and 36 places to eat or drink.